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Questions tagged [sheaf-theory]

A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

5
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1answer
368 views

Existence of acyclic coverings for a given sheaf

Let $\mathcal{F}$ be a sheaf over $X$ and $\mathcal{U}=\{U_i\}_{i\in I}$ a covering of $X$. I say that $\mathcal{U}$ is acyclic for $\mathcal{F}$ if $H^k(U_{i_0 \ldots U_n}, \mathcal{F}|_{U_{i_0 \...
6
votes
1answer
922 views

Fiber product of sheaves

If one has a topological space $X$ and three presheaves resp. sheaves $F$ and $G$ and $H$ of abelian groups on it with morphisms of presheaves resp. sheaves $F\rightarrow H$, $G \rightarrow H$, then I ...
23
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2answers
3k views

tensor product of sheaves commutes with inverse image

Let $f : X \to Y$ be a morphism of ringed spaces and $\mathcal{M}$, $\mathcal{N}$ sheaves of $\mathcal{O}_Y$-modules. Then one has a canonical isomorphism $f^*(\mathcal{M} \otimes_{\mathcal{O}_Y} \...
4
votes
0answers
55 views

Computing Exts and projective covers

Let $\mathcal{P}$ be the category of perverse sheaves on $\mathbb{P}^1$ over the field $\mathbb{C}$, where strata are the point $Z = {0}$, and its complement $U$. Let $i: Z \rightarrow \mathbb{P}^1$ ...
1
vote
1answer
226 views

Is the presheaf of continuous functions on a topological space a “complete presheaf”?

Is the presheaf of continuous functions $f:A\rightarrow B$ from a topological space $A$ to another topological space $B$ a "complete presheaf"? Can't find this, anyone have a reference?
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3answers
1k views

Reference for “It is enough to specify a sheaf on a basis”?

The wikipedia article on sheaves says: It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. ...
3
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1answer
183 views

Obtaining a morphism of sheaves from a morphism of presheaves

I think I need some help on a problem about sheaf theory. Suppose that $f: X \rightarrow Y$ is a continuous map of topological spaces, $\mathscr{F}$ is a sheaf on $X$. Prove that the morphism of ...
3
votes
2answers
720 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?
5
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0answers
307 views

Locally free sheaves on locally ringed spaces

One can define the notion of a locally free sheaf (of finite rank) on any locally ringed space. If you restrict to the category of (noetherian?) schemes, this category is equivalent to the category ...
22
votes
7answers
5k views

Examples of surjective sheaf morphisms which are not surjective on sections

Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of ...
1
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2answers
257 views

Isomorphism in derived category of coherent sheaves

this question might seem a bit special, but it came up at a crucial point of a proof I read and so I would be very obliged if someone could explain this to me: given is a smooth projective variety $X$...
4
votes
1answer
206 views

On the definition of morphisms of ringed spaces

Suppose that $(X,\mathcal{O}_X),(Y,\mathcal{O}_Y)$ are two ringed spaces and $(f,f^{\sharp}):(X,\mathcal{O}_X)\longrightarrow(Y,\mathcal{O}_Y)$ is a morphism of these two ringed spaces.I wonder why we ...
68
votes
2answers
9k views

Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
2
votes
3answers
344 views

Sheafification and the unique decomposition of morphisms

My question is an attempt to understand the proof of lemma 6.17.3 (page 164) in the Stacks Project: http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf This lemma states: Let $F$ ...
5
votes
2answers
535 views

Lifting sheaves from the special fibre to the generic fibre

Let $R$ be a complete discrete valuation ring and let $S = \operatorname{Spec}(R)$. Let $\mathcal{X}\to S$ be a complete, regular, flat, connected $S$-scheme of finite type whose fibres are smooth, ...
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1answer
114 views

(pre)sheaves on path connected neigh

Presheaves are defined by using neighborhoods of a point. Is there a way to restrict this construction to path connected neighborhoods of points? What is the name of the object which assigns other ...
4
votes
2answers
963 views

Sheaf cohomology

Let $\mathcal{F}$ be a sheaf over, say, a paracompact differentiable manifold $M$. Then to compute the cohomology $H(M,\mathcal{F})$ of $\mathcal{F}$, we can use any acyclic resolution or use Cech ...
7
votes
1answer
906 views

etale space v. covering space

On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic ...
4
votes
1answer
219 views

Cohomological dimension of a non-compact interval

Iversen, in "Cohomology of Sheaves," proves a number of theorems about the cohomological dimension of a locally compact space. In particular, if $\mathcal{F}$ is any sheaf on $\mathbb{R}^1$, it is ...
33
votes
3answers
2k views

Failure of isomorphisms on stalks to arise from an isomorphism of sheaves

It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
21
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1answer
557 views

functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define ...
7
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1answer
1k views

Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root

I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my ...
6
votes
1answer
107 views

Criterion for local systems

Given a sheaf on say a manifold, such that all its stalks are isomorphic to a fixed finite dimensional vector-space. Is it true that the sheaf is a local system?
7
votes
2answers
691 views

Hartshorne exercise about sheaves on $\mathbb{P}^1$

I've been stuck on Exercise II.1.21(e) from Hartshorne's book for quite a while. It concerns the projective line $\mathbb{P}^1$ over an algebraically closed field $k$: write $\mathscr{H}$ for the ...
5
votes
0answers
529 views

Identifying isomorphic schemes

Suppose $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ are isomorphic as schemes. Then by definition there is an isomorphism of locally ringed spaces $(\psi, \psi^{\sharp}): (X, \mathcal{O}_X) \to (Y, \...
4
votes
1answer
538 views

presheaf as a colimit of representables

It is well-known that any presheaf (for simplicity say we're talking about presheaves of sets on a topological space $X$) is a colimit of representable presheaves in a canonical way. This has been ...
10
votes
2answers
1k views

Inverse Image as the left adjoint to pushforward

Assume $X$ and $Y$ are topological spaces, $f : X \to Y$ is a continuous map. Let ${\bf Sh}(X)$, ${\bf Sh}(Y)$ be the category of sheaves on $X$ and $Y$ respectively. Modulo existence issues we can ...
2
votes
1answer
483 views

Exact sequence of sheaves

Let $X$ be a scheme and let $Y$ be a closed subscheme with ideal sheaf $I$. Let $F$ be a coherent sheaf on $X$. Is the sequence $$ 0 \to I\otimes F \to F \to F \otimes O_Y \to 0 $$exact? This is ...
14
votes
2answers
975 views

Inverse Limit of Sheaves

It is well-known that if you have an inverse system of abelian groups $(A_n)$ (this works in several other nice categories) in which all the maps are surjective (or at least satisfy the Mittag-Leffler ...
2
votes
1answer
465 views

The difference between the flabby sheaf and the fine sheaf

As we know,the flabby sheaf and the fine sheaf are all acyclic,but does one imply another?I need some examples to distinguish them,who can help me?
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votes
4answers
4k views

What are the differences between a fiber bundle and a sheaf?

They are similar. Both contain a projection map and one can define sections, moreover the fiber of the fiber bundle is just like the stalk of the sheaf. But what are the differences between them? ...
2
votes
1answer
91 views

a function of a dependent type, a section, a sheaf

I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...
2
votes
0answers
154 views

Isomorphism of schemes and invertible sheaves

I have a question more about terminology than anything else: If $f:X\rightarrow Y$ is an isomorphism of schemes, then what does it mean to say "the invertible sheaf $\mathscr{L}$ on $X$ corresponds ...
3
votes
1answer
274 views

Pushing forward sheaves and the result on sheaf cohomology

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and let $\cal{F}$ be a sheaf on $X$. Is there an obvious map $H^\ast(Y,f_\ast \mathcal{F} ) \rightarrow H^\ast (X,\cal{F})$?
4
votes
1answer
187 views

Is a presheaf which is a direct summand of a sheaf necessarily a sheaf?

Let's say $X$ is a topological space and let's consider the categories of sheaves and presheaves of abelian groups on $X$. Suppose we have a presheaf that is a direct summand (in the category of ...
8
votes
1answer
1k views

Global Sections of Sheaf and Dual

Maybe this is well-known, but suppose you have an invertible sheaf $\mathcal{L}$ on a scheme $X$. If $X$ is a projective space (or even projective bundle over an integral scheme, by a similar argument....
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votes
3answers
3k views

Tensor product of invertible sheaves

Given two invertible sheaves $\mathcal{F}$ and $\mathcal{G}$, one can define their tensor product, but in this definition $\mathcal{F} \otimes \mathcal{G} (U)$ is (apparently) not simply equal to $\...
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votes
2answers
2k views

Stalks of the tensor product presheaf of two sheaves

Let $(X, \mathscr{O})$ be a ringed space and $\mathscr{F}, \mathscr{G}$ be sheaves of $\mathscr{O}$-modules on $X$. Define $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$. ...
23
votes
3answers
2k views

Why doesn't Hom commute with taking stalks?

I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, G)(U)=Mor(...
5
votes
1answer
272 views

Subbundles and subsheaves

Let Let $E \rightarrow X$ be a vector bundle on a manifold $X$. Let $\cal E$ be the sheaf of sections of $E$. Let $\cal F$ be a subsheaf of $\cal E$, and let $F$ be the etale space of $\cal F$. What ...
5
votes
3answers
297 views

Can every element in the stalk be represented by a section in the top space?

Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map?
67
votes
0answers
5k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
12
votes
3answers
755 views

Computing stalks: do direct limits behave like limits?

Suppose that $X$ is a topological space with a sheaf of rings $\mathcal{O}_X$. In general, the stalk at a point $p \in X$ is the direct limit of the rings $\mathcal{O}_X(U)$ for all open sets $U$ ...